e ISSN–0976–7959 Asian Volume 11 | Issue 1 | June, 2016 | 78-86 DOI : 10.15740/HAS/AS/11.1/78-86 Science Visit us | www.researchjournal.co.in A CASE STUDY Mathematics in ancient India NIRANJAN SWAROOP Department of Mathematics, Christ Church College, KANPUR (U.P.) INDIA (Email: [email protected]) ABSTRACT “India was the motherland of our race and Sanskrit the mother of Europe’s languages. India was the mother of our philosophy, of much of our mathematics, of the ideals embodied in Christianity... of self-government and democracy. In many ways, Mother India is the mother of us all.”- Will Durant (American Historian 1885-1981). Mathematics had and is playing a very significant role in development of Indian culture and tradition. Mathematics in Ancient India or what we call it Vedic Mathematics had begun early Iron Age. The SHATAPATA BRAHAMANA AND THE SULABHSUTRAS includes concepts of irrational numbers, prime numbers, rule of three and square roots and had also solved many complex problems. Ancient Hindus have made tremendous contributions to the world of civilization in various fields such as Mathematics, Medicine, Astronomy, Navigation, Botany, Metallurgy, Civil Engineering, and Science of consciousness which includes Psychology and philosophy of Yoga. The object of this paper tries is to make society aware about the glorious past of our ancient science which has been until now being neglected without any fault of it. Key Words : Ancient, Mathematics, Ganita, Sulabhsutras View point paper : Swaroop, Niranjan (2016). Mathematics in ancient India. Asian Sci., 11 (1): 78-86, DOI : 10.15740/HAS/AS/11.1/78- 86. ny account of the classical sciences of India has wrote:- to begin with mathematics, the reasons being, “Occidental mathematics has in past centuries Athe ancient Sanskrit text Vedanga Jyotisa (ca. broken away from the Greek view and followed a course fourth century B.C.E.) says, which seems to have originated in India” Like the crest on the peacock’s head, Like the gem The love affair between Indian culture and numbers in the cobra’s hood so stands mathematics at the head has been long. When most societies had difficulty in of all the sciences. handling numbers beyond 1000, the Buddhist text Lalita- The Sanskrit word used for mathematics in this vistara (before the fourth century C.E.) not only has no verse is ganita, which literally means “reckoning.” problem with huge numbers, but seems to revel in giving The unique Indian view about the classical of them names (10145, the highest number quoted, being mathematics is that number was treated as the primary called dhvaja-nis’a-mani.) concept. Indian mathematics which we call today Ancient This thing has been substantiated by a distinguished Indian Mathematics has it’s origination in Indian Swiss mathematician-physicist in 1929 that when he subcontinent from 1200 BCE upto the end of the 18th HIND INSTITUTE OF SCIENCE AND TECHNOLOGY NIRANJAN SWAROOP century. In this classical period of Indian mathematics Ancient Indian was aware of the concept of zero, (400 CE to 1600 CE), not only important but excellent actually they are solely responsible for hugely important contributions were made by scholars such as invention in mathematics and it would be no hyperbole Aryabhata, Brahmagupta, Mahâvîra, Bhaskara II, to say that actually they invented the zero. Madhava of Sangamagrama and Nilakantha Somayaji. The earliest recorded date an inscription of zero on The decimal number system which we use today Sankheda Copper Plate was found in Gujrat In 585- was first recorded in Indian mathematics. Indian 586CE , not only this but also use of a circle which mathematicians made early contributions to the study of symbolizes the number zero is found in a 9th Century the concept of zero as a number, negative numbers, engraving in a temple in Gwalior in central India. But arithmetic, and algebra not only this but In addition, the brilliant conceptual leap to include zero as a number trigonometry was further advanced in India and in in its own right (not merely as a placeholder, a blank or particular, the modern definitions of sine and cosine were empty space within a number, as it had been treated developed here also in India. until that time) is usually associated to the 7th Century These mathematical concepts were transmitted to Indian mathematicians Brahmagupta - or possibly the Middle East, China, and Europe and led to further another Indian, Bhaskara I - even though it may well developments that now form the foundations of many have been in practical use for centuries before that. The areas of mathematics. use of zero as a number which could be used in Ancient and medieval Indian mathematical works, calculations and mathematical investigations would was composed in Sanskrit, which usually consisted of a revolutionize mathematics. section of sutras in which a set of rules or problems were Had there been no zero there would have been no stated with great beauty in verse in order to aid binary number system and as a result no computers and memorization by a student. This was followed by a how cumbersome would have been counting? second section consisting of a prose commentary The Hindu culture has positional number system in (sometimes multiple commentaries by different scholars) base ten they used a dot to represent an empty place, that explained the problem in more detail and provided sunya which meant empty was the name for this dot at justification for the solution. In the prose section, the this point earlier zero was a place holder and an aid in form (and therefore, its memorization) was not computation, by 500 C.E. the Hindus use a small circle considered as important as the ideas involved. to represent zero this circle was later on recognized as a All mathematical works were transmitted from one numeral zero. generation to other orally until approximately 500 BCE; thereafter, they were transmitted both orally and in The ancient Hindu symbol of a circle with a dot manuscript form. The oldest extant mathematical in the middle, known as bindu or bindhu, document produced on the Indian subcontinent is the symbolizing the void and the negation of the self, was probably instrumental in the use of a circle birch bark Bakhshali Manuscript, discovered in 1881 in as a representation of the concept of zero. the village of Bakhshali, near Peshawar (modern day Pakistan) and is likely from the 7th century CE. Fig. 2 : Bindu Zero in India The number 0 appear in the late 10th century in India. The concept of zero as a number and not merely a symbol for separation is attributed to India, where by the 9th century CE, practical calculation were carried out using zero, which was treated like any other number, even in case of division. The word for zero in Hindu-India is ‘Shunya” meaning void or empty. India is recognized with great respect for its invention of Zero by importance with technological world. These are the numbers that the Hindus and the Arabics use. Fig. 3 : Ancient Indian text explains concept of Infinity, as reciprocal of zero as well as they were also aware of Fig. 1 : The case of the “Pythagoras theorem” the concept of trigonometric functions Asian Sci., 11(1) June, 2016 : 78-86 79 HIND INSTITUTE OF SCIENCE AND TECHNOLOGY MATHEMATICS IN ANCIENT INDIA As x gets bigger and bigger 1 approaches zero. Conversely, as x gets smaller and The approximation formula : x smaller and approaches zero, 1 approaches infinity x The formula is given in verses 17 – 19, Chapter VII, Mahabhaskariya of Bhaskara I. A translation of the 1 5 x verses is given below: 4 (Now) I briefly state the rule (for finding 3 the bhujaphala and the kotiphala, etc.) without making 2 use of the Rsine-differences 225, etc. Subtract the 1 degrees of a bhuja (or koti) from the degrees of a half 0 circle (that is, 180 degrees). Then multiply the remainder -1 by the degrees of the bhuja or koti and put down the -2 result at two places. At one place subtract the result -3 from 40500. By one-fourth of the remainder (thus, obtained), divide the result at the other place as multiplied -4 anthyaphala (that is, the epicyclic radius). Thus, -5 by the ‘ is obtained the entirebahuphala (or, kotiphala) for the -5 -4 -4 -2 -1 0 1 2 3 4 5 sun, moon or the star-planets. So also are obtained the direct and inverse Rsines. (The reference “Rsine-differences 225” is an allusion to Aryabhata’s sine table.) In modern mathematical notations, for an angle x in degrees, this formula gives : 4x (180 – x) sin x0 40500 – x (180 – x) Equivalent forms of the formula : Bhaskara I’s sine approximation formula can be expressed using the radian measure of angles as follows (O’Connor and Robertson, 2000) : 16x( – x) sin x 2 5 – 4x( – x) For a positive integer n this takes the following form (George, 2009) : Fig. 4 : Sine approximation formula 16(n – 1) sin n 5n2 – 4n 4 In mathematics, Bhaskara I’s sine approximation formula is a rational expression in one variable for Equivalent forms of Bhaskara I’s formula have been the computation of the approximate values of given by almost all subsequent astronomers and the trigonometric sines discovered byBhaskara I (c. 600 mathematicians of India. For example, Brahmagupta’s – c. 680), a seventh-century Indian mathematician. This (598 – 668 CE) Brhma-Sphuta-Siddhanta (verses 23 formula is given in his treatise titled Mahabhaskariya. – 24, Chapter XIV) (Gupta, 1967) gives the formula in It is not known how Bhaskara I arrived at his the following form: approximation formula.